Well, to begin with, i´m spanish so sorry for my english.
I´ve made this guide because there aren´t good guides for KARTHUS and we need a really good one.
Also, on the Ability Power part, it says 580 AP, but the page doesnt add the extra 35% AP we have so we will end with more than 800 AP!!!!!! PLEASE READ THE WHOLE GUIDE. IT´S NOT VERY LONG AND ITS QUITE USEFUL!!! trust me :)
I´ve been playing with lots of AP(Ability power) champions(LeBlanc, Karthus, Kassadin), and I think Karthus is the most powerful of them.
I´m gonna explain you why.
Also you have to realise that the most powerful ability of Karthus is his LAY WASTE, so you need to use it as much as you can. If you really think that using LAY WASTE is very important, you will understand perfectly this guide and you will win most of the battles.

Karthus have a weak beggining so we need to make him stronger. That´s why we have selected the Archmague and Mental Force Masteries, so we have an extra 5% AP and 6 more AP.
Adding those masteries to our runes, we well have moreless 40 AP on the begining( 15 ap from the Quintessences, 15 from the runes (thats 30 ap), 6 from the masteries( 36 ap), and an extra 5%).
Also we start with an extra magic penetration for damaging mor the enemy champions in early game.

Now we have made the PRE-GAME, lets start the fight.

We are going to buy our first items.

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First of all we are going to buy 2x FAERIE CHARM, because as i said in the intro, LAY WASTE is very important on Karthus, so we need a high mana recover in the begining.
Also we are gonna buy a few health potions(x3).

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When we have farmed a bit and killed one champion( our mid enemy), we should have enough for buying ATHENE´S UNHOLY GRAIL, a very strong item that gives us 60 AP (WE ARE HAVING 100 AP AND WE ARE ONLY STARTING THE GAME), 40 magic resistance( 70 in total) and a very high mara recover with an extra recover each 1% mana we dont have( WITH A 50 % OF MANA WE SILL BE RECOVERING LIKE 8 PER SECOND). Also, we will have a 20% cooldown so we will be using our "R" more frecuently.
---- very important--- REMEMBER WE ARE BUILDING OUR KARTHUS FOR USING LAY WASTE, SO DONT TRY TO SAVE MANA... IT WIL BE RESTORED VERY QUICKLY!!!!

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Once finished it, we will buil our RoA( Rod of ages) and we will increase our AP in 80( 180 AP), 450 extra HP and 450 mana.

Well, our karthus is starting to be strong, but not as much as he can.

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Now we are going to build our Rabadon´s Cap[... it will increase our AP in 120+ 30 % of accumulated( i will write "(original AP+ the Rabadon´s percentage)"), so we will get 150 of rabadon´s cap(120+30), 80 of Athene´unholy grial( 60+20), and 100 of RoA( 80+20). in total... 330 AP!!! , but adding our extra 5% of the masteries and runes, we will have moreless 400 AP (moreless)...
we should be moreless in level 13, 14, in a normal game.

As i said, we are having a 20% of cooldown, very important for using our "R", so we are going to increase this cooldown in a 10% with the "DEATHFIRE GRASP". Its a very powerful item, personally, i dont use his active, because im very concentrated on using my "Q", but if you want, you can use it; it adds 150 AP( 120+30) and an extra 10 % cooldown; a total of 30% cooldown (it´s a minute less on our "R"), and ... 550 AP!!!!

OKAYYY it´s getting to ve a very high AP champion, but we haven´t ended yet.
We are gonna buy a Void Staff for increasing our AP penetration in a 35%( it´s not really a incrementation of our penetration, its a reduction of the enemy´s magic defense) so we will be very very close to the true damage. also it will increase our AP in 100 ( 70 + 30 moreless) so we are gonna have 640 AP!!! with a very high penetration, so the true damage is very close!!!

And now, the real end for our enemies... the Zhonya´s hourglass... it will increase our defense in 50, so we will be protected from Attack Damage Carriers (ADC), and our AP in 150 ( 120+30)

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[u]Summing up, we will have more than... 800 AP!!!, 30% cooldown, 70 magic defense, 100 attack defense, a really high mana recover( you will never run out of mana), and 35% reduction of enemy´s magic defense.[/u]

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WELL LET´S CONTINUE WITH THE SKILLS:

-DEATH DEFIED(PASSIVE) when dead, u can still use your skills for 7 seconds, but you can´t move.

-LAY WASTE "Q"
as i´ve said, it is the bread and the butter of Karthus. We can save distances with other champios using this spell and we can farm easily and safely. Use it as much as you can. you will have lots of mana so dont be stupid and **** them all!!!

-WALL OF PAIN "W"
THIS is one of the most useful skills. You can make sotp persecuting you a WHOLE FULL ENEMY TEAM!!! it reduces their speed an 80 % and their magic defenses in a 50 %( with the 35 % of the VOID STAFF, = -85 % magic defense) so you can escape easily
but the most useful use of "W", is to persecute enemies. when reduced, u go glose to them with your "E" activated so every second you damage them 250 HP!!!! EVERY SECOND!!!

-DEFILE "E"
very useful for persecuting enemies or fighting in a close combat, and for recovering extra mana with every criature killed( 44 mana)<-- PASSIVE. When activated, it creates a mid range area which damages 250 HP to everyone inside it(enemy´s criatures and champions)

-REQUIEM "R" 180 secs
most of the kills you are gonna do, will be with your ulti. Thats why we have an extra 30% cooldown (118 secs in total, so we have reduced it in a minute).
it damages 1000 AP to every champion in the map ( enemy, of course) so when you use it, you will kill one or two enemis. You cannot use it whenever u want to, you muste look to the health of every champion on the map, and if you really think you can kill him, use it.

1) USE YOUR "Q" EVERYTIME!!! in the highest level, u will cause to a single enemy 600 HP!!!

2)This is THE COMBO!!!
"W"+"E"+"Q": if someone is gonna atack you, place the wall of pain between him and you, so her magic def. will be lower and he will reduce his speed in a 80 %.
Next, activate your "E", so every second you will cause him 250 damage points (how many seconds could he stay inside it??? no more than 3 or 4).
finally atack him with your "Q". If he is alone, every "Q" will cause 600!! damage points, and you use it every 1,5 secs!!
LETS CALCULATE: 250x4=1000
600x3=1800
total=2800... dead
If he starts running away, use FLASH and GHOST, and in a few mor seconds, just by being near to him, he will die.

3) THE WAY FOR MULTI-KILLING
using your "R". everytime you see that an enemy champion is escaping from a fight with less than 1000 HP, use it. You will kill one or two enemies every 2 minutes... ONLY FOR STAYING IN YOUR BASE OR WHERE EVER YOU USE IT!! they will be sacred to death every time they are under 1000 HP because the know they could die soon!!!!

Thank you very much for reading!!!
I hope it can help you to be a multikill Karthus and for being respected by rivals and mates.
PLEASE LIKE IT , ITS JUST A CLICK!!!! AND COMMENT IF IT HAS HELPED YOU :)
SEE YOU!!